I would like to tell you a little bit about the SIAM 100-Digit Challenge. The challenge originated in a problem solving course taught by Lloyd N. Trefethen for incoming D.Phil students in numerical analysis at Oxford University. Each week a problem was given with no hints and the students had to find as many digits of the answer as they could. The reason Trefethen made the challenge public was to allow other mathematicians, especially numerical analysts, to have fun solving difficult problems. When Trefethen approached SIAM in 2001 with the idea of publishing 10 challenging scientist computation problems to the public, they enjoyed the idea so much that that they launched the contest in the January/February issue of SIAM News. The full text of Trefethen’s challenge is as follows [1]:
Each October, a few new graduate students arrive in Oxford to begin research for a doctorate in numerical analysis. In their first term, working in pairs, they take an informal course called the “Problem Solving Squad.” Each week for six weeks, I give them a problem, stated in a sentence or two, whose answer is a single real number. Their mission is to compute that number to as many digits of precision as they can.
Ten of these problems appear below. I would like to offer them as a challenge to the SIAM community. Can you solve them?
I will give $100 to the individual or team that delivers to me the most accurate set of numerical answers to these problems before May 20, 2002. With your solutions, send in a few sentences or programs or plots so I can tell how you got them. Scoring will be simple: You get a point for each correct digit, up to ten for each problem, so the maximum score is 100 points.
Fine print? You are free to get ideas and advice from friends and literature far and wide, but any team that enters the contest should have no more than half a dozen core members. Contestants must assure me that they have received no help from students at Oxford or anyone else who has already seen these problems.
Hint: They’re hard! If anyone gets 50 digits in total, I will be impressed. The ten magic numbers will be published in the July/August issue of SIAMNews, together with the names of winners and strong runners-up.
—Nick Trefethen, Oxford University.
After four months, the deadline arrived and entries were submitted by 94 teams from 25 countries with a total of 180 contestants. Among all the teams, 20 teams had perfect scores of 100 and 5 additional teams with scores of 99. Among the software systems that were used besides Mathematica, Maple, and MATLAB include: C, C++, Fortran, Java, Visual Basic, Turbo-Pascal, GMP, GSL, Octave, and Pari/GP. A book by Bornemann, Laruie, Wagon, and Waldvogel [2], four individuals that participated in the challenge, provides a large source of information about this challenge. The solutions they present are a synthesis of their own solutions and those of other participants.
In the December 2002 issue of SIAM News Joseph Keller of Stanford University published an interesting letter describing how few of the solutions presented verifiably correct solutions.
I found it surprising that no proof of the correctness of the answers was given. Omitting such proofs is the accepted procedure in scientific computing. However, in a contest for calculating precise digits, one might have hoped for more.
Proofs of correctness in numerical analysis take on many forms. An interesting numerical method that provides an algorithm and proof of correctness uses a branch of mathematics called interval analysis. Many of the solutions presented in [2] use interval analysis to prove the solutions submitted to the challenge are correct.
References
- Lloyd Trefethen. A Hundred-Dollar, Hundred-Digit challenge. SIAM News, 35 (1), 2002. URL http://www.siam.org/news/news.php?id=388.
- Folkmar Bornemann, Dick Laruie, Stan Wagon, and Jörg Waldvogel. The SIAM100-Digit Challenge: A Study in High-Accuracy Numerical Computing. SIAM, 2004.
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